Generally, people do take R and B to be fixed amounts of money—then, it’s worth pointing out that the expected value of the blue envelope is not .5*.5R+.5*2R, because R there stands for 2 different amounts of money.
If the amounts of money are $5 and $10, then R in the first case stood for $10 and R in the second case stood for $5, so it should really be .5*.5($10)+.5*2($5), and the expected value is just $7.50.
Generally, people do take
RandBto be fixed amounts of money—then, it’s worth pointing out that the expected value of the blue envelope is not.5*.5R+.5*2R, becauseRthere stands for 2 different amounts of money.If the amounts of money are
$5and$10, thenRin the first case stood for$10andRin the second case stood for$5, so it should really be.5*.5($10)+.5*2($5), and the expected value is just$7.50.Yes, I like your simplification.
To amplify it and make the results explicit, Suppose my envelope strategy is to flip a fair coin and
heads: R=$5 B=$10
tails: R=$10 B=$5.
Then
E(R) = E(B) = $7.50
E(B)/E(R) = E(B)/E(R) =1
E(R/B) = E(B/R) = 1.25